Problem: Simplify the following expression: $y = \dfrac{-5x^2- 12x- 7}{x + 1}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(-7)} &=& 35 \\ {a} + {b} &=& &=& {-12} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $35$ and add them together. The factors that add up to ${-12}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${-5}$ $ \begin{eqnarray} {ab} &=& ({-7})({-5}) &=& 35 \\ {a} + {b} &=& {-7} + {-5} &=& -12 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-5}x^2 {-7}x) + ({-5}x {-7}) $ Factor out the common factors: $ x(-5x - 7) + 1(-5x - 7)$ Now factor out $(-5x - 7)$ $ (-5x - 7)(x + 1)$ The original expression can therefore be written: $ \dfrac{(-5x - 7)(x + 1)}{x + 1}$ We are dividing by $x + 1$ , so $x + 1 \neq 0$ Therefore, $x \neq -1$ This leaves us with $-5x - 7; x \neq -1$.